Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 18 + 6\cdot 61 + 16\cdot 61^{2} + 53\cdot 61^{3} + 4\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 + 10\cdot 61 + 14\cdot 61^{2} + 4\cdot 61^{3} + 26\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 + 21\cdot 61 + 53\cdot 61^{2} + 15\cdot 61^{3} + 39\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 17\cdot 61 + 3\cdot 61^{2} + 41\cdot 61^{3} + 3\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 + 41\cdot 61 + 61^{2} + 33\cdot 61^{3} + 23\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 + 27\cdot 61 + 53\cdot 61^{2} + 46\cdot 61^{3} + 46\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 46 + 37\cdot 61 + 5\cdot 61^{2} + 41\cdot 61^{3} + 22\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 53 + 20\cdot 61 + 35\cdot 61^{2} + 8\cdot 61^{3} + 16\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,4,5)(2,8,7,6)$ |
| $(1,2)(3,6)(4,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,7)(3,5)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,6)(4,7)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,3)(4,6)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,3,4,5)(2,8,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
